It certainly can, but the impact it will have on your tuning depends on a lot of factors:

How old your strings are

What type and weight strings you use

How solid your tuners are -- locking tuners will stay in tune better

How deep you're dipping the whammy bar; big dive-bombs will put you out of tune faster

How well you've stretched/broken in your strings if you've recently re-strung the guitar -- doing a few deep bends can help keep the tuning more stable while you're playing

One thing that helped me keep tune with use of a vibrato (both Fender-style and Bigsby) is using a tuning lubricant at the nut and saddle. Particularly with the wound strings, a deep bend can cause the strings to slide a little bit over the nut and/or saddle, and if they can't slide easily back into position, this might put you out of tune.

I personally like "Big Bends Nut Sauce" tuning lubricant, but I haven't done a lot of comparison. It was recommended to me by a local luthier. You put just a tiny but in each groove in the nut and each saddle when you re-string, and it really helps keep the guitar in tune (even if you don't have a vibrato -- it still helps keep in tune through deep bends and such.)

Careful setup of the guitar will help to minimise the effect that it can have. As mentioned by other answers, a good double locking unit will reduce the problems drastically, but even a fender style can work pretty well if you put the time and effort in. Check out this vid